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The experiment: use the Φ-corkscrew system to search its own manifold
for the direction that maximizes compression ratio. 5 domain experts
designed their components in parallel.
EXPERIMENT: EXPERIMENT_RADIAL_SELF_FIND.md
- Hypothesis: ∃ d* on S⁷: walking γ_{d*} monotonically increases C(n)
- Method: Self-referential geodesic search with radial exploration
- Predictions: gradient exists, ascent converges, self-encoding helps
AGENT 1 — GeometricPhysicist: experiment_geodesic_search.md
- Geodesic: γ_d(t) = cos(t)·x + sin(t)·d (great circles on S⁷)
- Gradient ascent: exponential map + parallel transport
- Direction sampling: uniform, Φ-guided, gradient-biased
- 3 core functions: geodesic_search, gradient_ascent_step, sample_directions
AGENT 2 — InformationTheorist: experiment_compression_metric.md
- C(n) = L_S / |RLE(DNA(phinary(n)))|
- Bounds: Ω(L_S/log n) ≤ C(n) ≤ O(L_S/log log n)
- Key insight: phinary constraint inherently favors compressibility
- Entropy H(n), Kolmogorov K(n), spectral radius analysis
AGENT 3 — SystemsEngineer: experiment_feedback_loop.md (2,033 lines!)
- 12-state, 15-transition state machine
- 3-layer strange loop containment (bounded, contractive, depth cap)
- Radial exploration: OUTWARD/INWARD/OSCILLATE modes
- Full FAMM-DAG integration with meltdown recovery
- 7 convergence criteria
AGENT 4 — FormalVerifier: experiment_formal_verification.md
- 8 Lean 4 theorems + master theorem
- Key: Bijection Preservation (search transform preserves injectivity)
- Paradox Prevention theorem (self-referential safety)
- 10 invariants, 5 verification conditions
- Integrates with ChentsovFinite.lean, quine.py proofs
AGENT 5 — MetaMathematician: experiment_meta_analysis.md
- Strange loop converges (C(n) is Lyapunov function, S⁷ compact)
- Fixed points exist (Brouwer + Kleene recursion theorem)
- Gödel boundary is epistemological, not ontological
- System finds itself but cannot prove global optimality
- 12 formal theorems
Total: 6 files, ~6,000 lines of experiment design
Refs: PHI_CORKSCREW_PERFECT_RECOVERY.md, PROOF_SELFSIGHT.md,
ChentsovFinite.lean, GoldenSpiralManifold.lean
95 KiB
95 KiB
SELF-FINDING FEEDBACK LOOP — Complete System Design
Radial Self-Finding Experiment: Φ-Corkscrew Searching Its Own Manifold
1. SYSTEM OVERVIEW
1.1 The Core Loop
┌─────────────────────────────────────────────────────────────────────────────────┐
│ SELF-FINDING FEEDBACK LOOP (SFFL) │
│ │
│ ┌──────────┐ ┌──────────┐ ┌──────────┐ ┌──────────┐ │
│ │ STATE │────>│ EXPLORE │────>│ EVALUATE │────>│ UPDATE │──┐ │
│ │ S_k │ │ N dirs │ │ C(n) │ │ d* , S* │ │ │
│ └────┬─────┘ └──────────┘ └──────────┘ └────┬─────┘ │ │
│ ↑ │ │ │
│ │ ┌──────────┐ ┌──────────┐ │ │ │
│ └────────────│ ENCODE │<────│ META │<─────────┘ │ │
│ │ SELF │ │ DECIDE │ │ │
│ └────┬─────┘ └────┬─────┘ │ │
│ │ │ │ │
│ ▼ ▼ │ │
│ ┌──────────────────────────┐ │ │
│ │ STRANGE LOOP CONTAINER │ │ │
│ │ (trajectory ──> next S) │─────────────────────┘ │
│ └──────────────────────────┘ │
│ │
│ ┌─────────────────────────────────────────────────────────────────────────┐ │
│ │ INNER LOOP (per iteration): │ │
│ │ State → Explore N directions → Evaluate compression → Pick best │ │
│ │ → Meta-decision → Encode trajectory → New state │ │
│ │ │ │
│ │ OUTER LOOP (convergence): │ │
│ │ Repeat inner loop until: │ │
│ │ - C plateaus (no improvement > ε for K iterations) │ │
│ │ - Gradient vanishes (||∇C|| < δ) │ │
│ │ - Max iterations reached (safety) │ │
│ │ - Meltdown detected (Baker-analogue violation) │ │
│ │ │ │
│ │ STRANGE LOOP (self-reference): │ │
│ │ The search trajectory T_k = {(d_i, t_j, C_ij)} becomes │ │
│ │ encoded as n_exp = spiral_index(T_k) and FED BACK as the │ │
│ │ starting state for the next iteration. │ │
│ │ │ │
│ │ This is NOT infinite regress because: │ │
│ │ - Trajectory is BOUNDED (finite N x M samples) │ │
│ │ - Encoding is CONTRACTIVE (C(n_exp) ≤ C(n_k) guaranteed) │ │
│ │ - Depth is CAPPED (max_self_ref_depth = D) │ │
│ └─────────────────────────────────────────────────────────────────────────┘ │
└─────────────────────────────────────────────────────────────────────────────────┘
1.2 Integration with FAMM + DAG + DNA
SFFL sits ON TOP of the co-evolution stack:
┌─────────────────────────────────────────┐
│ SELF-FINDING FEEDBACK LOOP (this doc) │
│ - State machine for the experiment │
│ - Strange loop containment │
│ - Convergence orchestration │
├─────────────────────────────────────────┤
│ CO-EVOLUTION ENGINE (COEVOLUTION_MODEL) │
│ - DAG: ResumableDAG checkpoints │
│ - FAMM: delay-line memory + scars │
│ - FSDU: scar computation │
│ - DNA: re-encoding + sort │
├─────────────────────────────────────────┤
│ BAKER-ANALOGUE (FAMM_BAKER_ANALOGUE) │
│ - Invariant: |Λ| ≥ ε OR Ω > 0 │
│ - Gate: admit/scar/reject │
│ - Scar pressure field │
├─────────────────────────────────────────┤
│ MANIFOLD LAYER (STATE_SPACE_EMBEDDING) │
│ - Δ₇: Hachimoji simplex │
│ - S⁷: Fisher sphere in √p-coords │
│ - g = g_Δ ⊕ g_FAMM ⊕ g_scar │
├─────────────────────────────────────────┤
│ ENCODING LAYER (SMUGGLE_MODEL) │
│ - Φ-corkscrew: f(n) = (√n·cos(nψ), │
│ √n·sin(nψ)) │
│ - DNA: RLE(phinary(n)) in 8 bases │
│ - Compression: C(n) = orig / compressed │
└─────────────────────────────────────────┘
2. STATE MACHINE
2.1 States
┌──────────────────────────────────────────────────────────────────────────────┐
│ SFFL STATE MACHINE │
│ │
│ ┌─────────┐ init ┌─────────┐ │
│ │ IDLE │────────────>│ INIT │ │
│ └─────────┘ └────┬────┘ │
│ │ seed RNG, validate S_0 │
│ ▼ │
│ ┌─────────────────────────────────────────┐ │
│ │ ┌─────────┐ fail ┌─────────┐ │ │
│ │ │ │────────────>│ RECOVER │ │ │
│ │ │ EXPLORE │────────────>│ │ │ │
│ │ │ │<────────────│ │ │ │
│ │ └────┬────┘ resume └─────────┘ │ │
│ │ │ explore N directions │ │
│ │ ▼ │ │
│ │ ┌─────────┐ fail ┌─────────┐ │ │
│ │ │ │────────────>│ RECOVER │ │ │
│ │ │EVALUATE │────────────>│ │ │ │
│ │ │ │<────────────│ │ │ │
│ │ └────┬────┘ resume └─────────┘ │ │
│ │ │ measure C(n) for each │ INNER LOOP │
│ │ ▼ │ (per iteration) │
│ │ ┌─────────┐ fail ┌─────────┐ │ │
│ │ │ │────────────>│ RECOVER │ │ │
│ │ │ SELECT │────────────>│ │ │ │
│ │ │ BEST │<────────────│ │ │ │
│ │ └────┬────┘ resume └─────────┘ │ │
│ │ │ find d*, t*, C* │ │
│ │ ▼ │ │
│ │ ┌─────────┐ fail ┌─────────┐ │ │
│ │ │ │────────────>│ RECOVER │ │ │
│ │ │ SELF- │────────────>│ │ │ │
│ │ │ ENCODE │<────────────│ │ │ │
│ │ └────┬────┘ resume └─────────┘ │ │
│ │ │ encode trajectory as n_exp │ │
│ │ ▼ │ │
│ │ ┌─────────┐ fail ┌─────────┐ │ │
│ │ │ │────────────>│ RECOVER │ │ │
│ │ │ META │────────────>│ │ │ │
│ │ │ DECIDE │<────────────│ │ │ │
│ │ └────┬────┘ resume └─────────┘ │ │
│ │ │ decide: ascend / stay / fail │ │
│ │ ▼ │ │
│ │ ┌─────────────────────────────────┐ │ │
│ │ │ Convergence check: │ │ │
│ │ │ - C plateau? ──> CONVERGED │ │ │
│ │ │ - Gradient < δ? ──> CONVERGED │ │ │
│ │ │ - Max iter? ──> HALTED │ │ │
│ │ │ - Meltdown? ──> PANIC │ │ │
│ │ │ - Otherwise ──> EXPLORE │ │ │
│ │ └─────────────────────────────────┘ │ │
│ └─────────────────────────────────────────┘ │
│ │
│ │
│ ┌─────────┐ converge ┌─────────┐ report ┌─────────┐ │
│ │ EXPLORE │───────────────>│CONVERGED│───────────────>│ REPORT │ │
│ └─────────┘ └─────────┘ └────┬────┘ │
│ │ │
│ ▼ │
│ ┌─────────┐ │
│ │ END │ │
│ └─────────┘ │
│ │
│ Any state ──meltdown──> PANIC ──unrecoverable──> END (with scar dump) │
│ │
└──────────────────────────────────────────────────────────────────────────────┘
2.2 State Definitions
| State | Description | Entry Condition | Exit Condition |
|---|---|---|---|
IDLE |
Pre-initialization | System start | init() called |
INIT |
Setup and validation | From IDLE |
S_0 validated, RNG seeded |
EXPLORE |
Generate N directions, walk geodesics | From INIT or META_DECIDE |
All directions explored |
EVALUATE |
Measure compression for each point | From EXPLORE |
All C(n) computed |
SELECT_BEST |
Find d*, t*, S* maximizing C | From EVALUATE |
Best direction identified |
SELF_ENCODE |
Encode trajectory as n_exp | From SELECT_BEST |
n_exp computed |
META_DECIDE |
Ascend, stay, or explore more | From SELF_ENCODE |
Decision made |
CONVERGED |
Local maximum found | Convergence criteria met | Final report |
HALTED |
Max iterations reached | Safety limit hit | Final report |
RECOVER |
Resume from DAG checkpoint | Any state fails | Recovered to last good |
PANIC |
Unrecoverable meltdown | Baker-analogue violated fatally | Scar dump + exit |
REPORT |
Emit final receipt | From CONVERGED or HALTED |
Done |
END |
Terminal | From REPORT or PANIC |
— |
2.3 State Transitions (Formal)
transition : State × Event → State
INIT × SeedValidated → EXPLORE
EXPLORE × DirectionsComplete → EVALUATE
EXPLORE × Meltdown → RECOVER
EVALUATE × CompressionDone → SELECT_BEST
EVALUATE × Meltdown → RECOVER
SELECT_BEST × BestFound → SELF_ENCODE
SELECT_BEST × NoImprovement → CONVERGED (if K consecutive)
SELF_ENCODE × Encoded → META_DECIDE
SELF_ENCODE × Meltdown → RECOVER
META_DECIDE × Ascend → EXPLORE (S_{k+1} = S*)
META_DECIDE × Stay → EXPLORE (S_{k+1} = S_self)
META_DECIDE × Converged → CONVERGED
META_DECIDE × MaxIterations → HALTED
META_DECIDE × Meltdown → RECOVER
RECOVER × ResumeSuccess → [previous state]
RECOVER × ResumeFail → PANIC
CONVERGED × ReportEmitted → REPORT
HALTED × ReportEmitted → REPORT
REPORT × Done → END
PANIC × ScarDumped → END
3. STATE VARIABLES — Complete Specification
3.1 Core State Vector
SFFLState = {
# ───────────────────────────────────────────────────────────
# 1. MANIFOLD POSITION (where we are on S⁷)
# ───────────────────────────────────────────────────────────
"current_point": {
"p": Vector8, # probability distribution on Δ₇
"x_sqrt": Vector8, # √p coordinates on S⁷ (||x|| = 1)
"n_spiral": uint64, # spiral index: closest point on Φ-corkscrew
"geodesic_origin": Vector8, # where this iteration started
},
# ───────────────────────────────────────────────────────────
# 2. SEARCH TRAJECTORY (the "strange loop" container)
# ───────────────────────────────────────────────────────────
"trajectory": {
"history": List[TrajectoryPoint], # all (d_i, t_j, C_ij) tuples
"self_ref_depth": uint8, # current recursion depth (0..D)
"encoded_trajectory": uint64, # n_exp = spiral_index(trajectory)
"trajectory_compression": float, # C(n_exp) — compression of the search
"cumulative_scar": ScarMeasure, # accumulated failed regions
},
# ───────────────────────────────────────────────────────────
# 3. CONVERGENCE TRACKING
# ───────────────────────────────────────────────────────────
"convergence": {
"C_history": Deque[float], # last K compression values
"gradient_estimate": Vector7, # ∇_d C (7D tangent to S⁷)
"gradient_norm_history": Deque[float],
"plateau_count": uint8, # iterations with |ΔC| < ε
"best_C": float, # global best compression
"best_n": uint64, # global best spiral index
"best_point": Vector8, # global best position on S⁷
"iteration": uint32, # current iteration count
},
# ───────────────────────────────────────────────────────────
# 4. RADIAL EXPLORATION STATE
# ───────────────────────────────────────────────────────────
"radial": {
"scale": float, # current radial scale (log₂ n)
"scale_history": Deque[float], # track scale changes
"radial_velocity": float, # d(scale)/dt — radial momentum
"radial_mode": Enum, # INWARD / OUTWARD / OSCILLATE
"arm_index": uint8, # which spiral arm (for multi-arm)
},
# ───────────────────────────────────────────────────────────
# 5. DETERMINISM & REPRODUCIBILITY
# ───────────────────────────────────────────────────────────
"determinism": {
"master_seed": uint64, # master RNG seed (immutable)
"iteration_seed": uint64, # seed for this iteration
"direction_seed": uint64, # seed for direction generation
"step_seed": uint64, # seed for step-size sampling
"rng_state": bytes, # full RNG state snapshot
},
# ───────────────────────────────────────────────────────────
# 6. FAMM + DAG INTEGRATION
# ───────────────────────────────────────────────────────────
"checkpoint": {
"dag_node_id": uint64, # current node in the DAG
"parent_node_id": Optional[uint64],# parent in DAG tree
"famm_cell_id": uint64, # FAMM cell storing this state
"transform_chain": List[Matrix], # T_1, T_2, ..., T_k transforms
"scar_field_hash": bytes32, # hash of accumulated scar field
},
# ───────────────────────────────────────────────────────────
# 7. EXPERIMENT METADATA
# ───────────────────────────────────────────────────────────
"meta": {
"experiment_id": str, # unique ID for this run
"start_time": timestamp, # wall-clock start
"last_checkpoint_time": timestamp, # for resume
"status": StateEnum, # current state machine state
"config": ExperimentConfig, # tunable parameters
},
}
3.2 TrajectoryPoint (the self-referential atom)
TrajectoryPoint = {
"iteration": uint32, # which iteration this point belongs to
"direction_index": uint16, # which of N directions
"direction": Vector8, # unit vector on tangent space T_{S⁷}
"step_index": uint16, # which step along geodesic
"step_size": float, # t_j: geodesic parameter
"point": Vector8, # γ_{d_i}(t_j) — actual point on S⁷
"spiral_index": uint64, # n_ij = spiral_index(point)
"compression": float, # C_ij = compression_ratio(n_ij)
"dna_encoding": str, # RLE(phinary(n_ij)) — the actual encoding
"encoding_size_bytes": uint32, # size of DNA encoding
"famm_gate_result": str, # "ADMIT" | "SCAR" | "REJECT"
"timestamp": uint64, # tick count when recorded
}
3.3 ExperimentConfig (Tunable Parameters)
ExperimentConfig = {
# Exploration
"N_directions": 32, # number of directions to explore per iteration
"M_steps_per_direction": 16, # steps along each geodesic
"T_max": 0.5, # max geodesic parameter (fraction of π)
"T_min": 0.01, # min geodesic step
# Radial exploration
"radial_enabled": True, # enable radial mode
"radial_scales": [0.5, 1.0, 2.0, 4.0], # log₂(n) scales to probe
"radial_momentum": 0.7, # velocity decay for radial mode
# Convergence
"K_plateau": 5, # iterations of |ΔC| < ε before plateau
"epsilon_plateau": 1e-6, # compression improvement threshold
"delta_gradient": 1e-8, # gradient norm threshold
"max_iterations": 1000, # hard stop
"max_wall_time_seconds": 3600, # wall-clock limit
# Self-reference containment
"max_self_ref_depth": 3, # max strange-loop nesting
"trajectory_budget": 10000, # max trajectory points before forced encode
"trajectory_compression_target": 0.5, # stop when C(n_exp) < target
# Determinism
"master_seed": 0xFEEDFACE42424242, # default master seed
# Checkpointing
"checkpoint_interval_iterations": 10, # checkpoint every N iterations
"checkpoint_interval_seconds": 300, # or every N seconds
"dag_max_branching": 4, # max children per DAG node
"famm_compression": True, # compress FAMM cells
# Baker-analogue (FAMM integration)
"famm_epsilon_factor": 1.0, # ε = factor * state_complexity
"famm_scar_pressure_decay": 0.95, # scar pressure decay per iteration
}
4. UPDATE RULES — Precise Pseudocode
4.1 INIT → EXPLORE
def initialize(config: ExperimentConfig) -> SFFLState:
"""Create initial state S_0 from configuration."""
# 1. Seed RNG hierarchy deterministically
master_seed = config.master_seed
rng = DeterministicRNG(master_seed)
# 2. Create initial point on S⁷
# Start at uniform distribution (center of simplex)
p_uniform = [1/8] * 8
x_sqrt = [sqrt(1/8)] * 8 # on S⁷: ||x||² = 8 * (1/8) = 1 ✓
# 3. Compute initial spiral index
n_0 = spiral_index(x_sqrt)
# 4. Measure initial compression
C_0 = compression_ratio(n_0)
# 5. Create initial DAG node
dag_node = dag.create_root_node(
point=x_sqrt,
spiral_index=n_0,
compression=C_0,
transform=IdentityTransform(),
)
# 6. Store initial FAMM cell
famm_cell = famm_bank.store(
data=pack_state(x_sqrt, n_0, C_0),
delay=f(1.0), # initial delay = neutral
delayMass=Tr(g_uniform), # trace of Fisher metric at uniform dist
delayWeight=1.0, # full coverage
)
# 7. Initialize scar field (empty)
scar_field = ScarMeasure.empty()
# 8. Construct state
state = SFFLState(
current_point=ManifoldPoint(
p=p_uniform,
x_sqrt=x_sqrt,
n_spiral=n_0,
geodesic_origin=x_sqrt,
),
trajectory=Trajectory(
history=[],
self_ref_depth=0,
encoded_trajectory=n_0,
trajectory_compression=C_0,
cumulative_scar=scar_field,
),
convergence=Convergence(
C_history=Deque([C_0], maxlen=config.K_plateau + 2),
gradient_estimate=zero_vector(7),
gradient_norm_history=Deque([inf], maxlen=config.K_plateau + 2),
plateau_count=0,
best_C=C_0,
best_n=n_0,
best_point=x_sqrt,
iteration=0,
),
radial=RadialState(
scale=log2(n_0 + 1),
scale_history=Deque(maxlen=10),
radial_velocity=0.0,
radial_mode=RadialMode.OSCILLATE,
arm_index=0,
),
determinism=Determinism(
master_seed=master_seed,
iteration_seed=hash(master_seed, 0),
direction_seed=hash(master_seed, 1),
step_seed=hash(master_seed, 2),
rng_state=rng.snapshot(),
),
checkpoint=Checkpoint(
dag_node_id=dag_node.id,
parent_node_id=None,
famm_cell_id=famm_cell.id,
transform_chain=[IdentityTransform()],
scar_field_hash=scar_field.hash(),
),
meta=Metadata(
experiment_id=generate_id(),
start_time=now(),
last_checkpoint_time=now(),
status=State.INIT,
config=config,
),
)
# 9. Persist initial checkpoint
dag_checkpoint(state)
state.meta.status = State.EXPLORE
return state
4.2 EXPLORE Phase (Direction Generation)
def explore(state: SFFLState) -> Tuple[List[Direction], SFFLState]:
"""Generate N deterministic directions from current point on S⁷."""
config = state.meta.config
rng = DeterministicRNG.restore(state.determinism.rng_state)
# ── DIRECTION GENERATION ─────────────────────────────────
# We generate directions using Φ-guided coverage for optimal
# exploration of the 7-sphere. The golden angle ψ ensures
# directions are maximally separated.
ψ = 2π / Φ² # golden angle ≈ 137.507° (for 7-sphere coverage)
N = config.N_directions
directions = []
# Seed for this iteration's directions
dir_seed = hash(state.determinism.master_seed, state.convergence.iteration, "directions")
dir_rng = DeterministicRNG(dir_seed)
for i in range(N):
# Generate direction using generalized Fibonacci / Φ-sequence
# on the 7-sphere. We use the Richtmyer sequence for
# quasi-random coverage of S⁷.
angles = []
for dim in range(7): # 7 angles parameterize S⁷
# Use Φ-based sequence for each dimension
angle = π * (dim + 1) * (i + 1) * ψ % (2π)
# Add small perturbation seeded per direction
perturbation = dir_rng.gaussian(0, 0.05)
angles.append((angle + perturbation) % (2π))
# Convert angles to unit vector on S⁷ (hyperspherical coords)
direction = angles_to_unit_vector(angles) # ||d|| = 1
# Ensure tangent to S⁷ at current point: project out radial component
x = state.current_point.x_sqrt
d_tangent = direction - dot(direction, x) * x
d_tangent = normalize(d_tangent) # ||d_tangent|| = 1, <d, x> = 0
directions.append(Direction(
index=i,
vector=d_tangent,
angles=angles,
seed=hash(dir_seed, i),
))
# ── RADIAL DIRECTIONS (if enabled) ──────────────────────
if config.radial_enabled:
# Add radial directions: fixed axes in spiral-index space
# These correspond to "go deeper" / "go shallower" on the spiral
radial_directions = generate_radial_directions(
current_n=state.current_point.n_spiral,
scales=config.radial_scales,
mode=state.radial.radial_mode,
)
directions.extend(radial_directions)
# ── SCAR AVOIDANCE ──────────────────────────────────────
# Filter directions that would enter scarred (failed) regions
# Uses the FAMM scar field accumulated from previous iterations
directions = scar_filter(directions, state.trajectory.cumulative_scar)
# Update state
state.determinism.direction_seed = dir_seed
state.determinism.rng_state = rng.snapshot()
return directions, state
def generate_radial_directions(current_n: uint64, scales: List[float],
mode: RadialMode) -> List[Direction]:
"""Generate directions that move along the spiral's radial dimension.
Unlike surface geodesics (which stay on S⁷), radial directions
change the spiral index n directly, which corresponds to changing
the "depth" or "scale" of the encoding.
"""
current_scale = log2(current_n + 1)
directions = []
for scale_delta in scales:
# Compute target n for this scale
target_scale = current_scale + scale_delta
target_n = int(2 ** target_scale)
# Direction in spiral-index space: (current_n → target_n)
# This maps to a direction on S⁷ via the spiral's inverse map
spiral_point_current = corkscrew(current_n) # f(n) = (√n·cos(nψ), √n·sin(nψ))
spiral_point_target = corkscrew(target_n)
# Direction on the disk that the spiral lives on
disk_direction = spiral_point_target - spiral_point_current
# Lift to S⁷: the radial direction "pushes" the point toward
# a different region of the spiral
direction = lift_to_S7(disk_direction)
directions.append(Direction(
index=1000 + len(directions), # offset to distinguish from surface dirs
vector=direction,
angles=None,
radial=True,
scale_delta=scale_delta,
target_n=target_n,
))
return directions
4.3 EVALUATE Phase (Compression Measurement)
def evaluate(state: SFFLState, directions: List[Direction])
-> Tuple[List[TrajectoryPoint], SFFLState]:
"""Walk geodesics along each direction and measure compression."""
config = state.meta.config
trajectory_points = []
x0 = state.current_point.x_sqrt
T_min = config.T_min
T_max = config.T_max
M = config.M_steps_per_direction
for direction in directions:
# Skip directions filtered by scar avoidance
if direction.skip:
continue
d = direction.vector
# Walk geodesic γ_d(t) from t=T_min to t=T_max
for j in range(M):
t = T_min + (T_max - T_min) * j / (M - 1)
# Compute geodesic on S⁷ from x0 in direction d
# γ_d(t) = cos(t)·x0 + sin(t)·d (great circle geodesic)
x_t = cos(t) * x0 + sin(t) * d
x_t = normalize(x_t) # stay on S⁷
# Map back to probability simplex
p_t = x_t ** 2 # element-wise square
p_t = p_t / sum(p_t) # normalize to probability
# Compute spiral index
n_t = spiral_index(x_t)
# Compute compression ratio
C_t = compression_ratio(n_t)
# Compute DNA encoding size
dna = RLE(phinary_encode(n_t))
encoding_size = len(dna) # in bases
# FAMM gate: Baker-analogue check
collapse = collapse_functional(state, p_t)
epsilon = famm_epsilon(state)
if abs(collapse) >= epsilon:
gate_result = "ADMIT" # Case I: rigidity
else:
# Record scar
scar = Scar(pressure=abs(collapse), mode="EXPLORE",
location=p_t, iteration=state.convergence.iteration)
state.trajectory.cumulative_scar.add(scar)
gate_result = "SCAR" # Case II: memory
point = TrajectoryPoint(
iteration=state.convergence.iteration,
direction_index=direction.index,
direction=d,
step_index=j,
step_size=t,
point=x_t,
spiral_index=n_t,
compression=C_t,
dna_encoding=dna,
encoding_size_bytes=encoding_size,
famm_gate_result=gate_result,
timestamp=tick(),
)
trajectory_points.append(point)
# Store in trajectory history
state.trajectory.history.extend(trajectory_points)
# Trim history if exceeds budget
if len(state.trajectory.history) > config.trajectory_budget:
# Compress: encode old history into a single meta-point
old_points = state.trajectory.history[:-config.trajectory_budget]
meta_point = encode_history_summary(old_points)
state.trajectory.history = [meta_point] + state.trajectory.history[-config.trajectory_budget:]
return trajectory_points, state
def compression_ratio(n: uint64) -> float:
"""C(n) = original_size / compressed_size"""
original_size = state.meta.config.original_size_bytes # e.g., 30GB
# Encode n in phinary, then RLE compress
phinary_str = phinary_encode(n)
dna_sequence = RLE(phinary_str)
# Each DNA base = 3 bits (8 symbols)
compressed_bits = len(dna_sequence) * 3
compressed_bytes = compressed_bits / 8
return original_size / compressed_bytes
4.4 SELECT_BEST Phase
def select_best(state: SFFLState, points: List[TrajectoryPoint])
-> Tuple[Vector8, uint64, float, Direction]:
"""Find the direction d* and step t* that maximize compression."""
if not points:
# No valid points found (all directions scarred)
raise ConvergenceException("No valid directions — all regions scarred")
# Find best point
best_point = max(points, key=lambda p: p.compression)
# Find best direction (the one containing the best point)
best_direction = Direction(
index=best_point.direction_index,
vector=best_point.direction,
)
# Update global best if improved
if best_point.compression > state.convergence.best_C:
state.convergence.best_C = best_point.compression
state.convergence.best_n = best_point.spiral_index
state.convergence.best_point = best_point.point
state.convergence.plateau_count = 0
else:
state.convergence.plateau_count += 1
# Estimate gradient
# Fit a quadratic to C vs t along each direction, estimate ∇C
gradient = estimate_gradient(points, state.current_point.x_sqrt)
state.convergence.gradient_estimate = gradient
state.convergence.gradient_norm_history.append(norm(gradient))
return (best_point.point, best_point.spiral_index,
best_point.compression, best_direction)
def estimate_gradient(points: List[TrajectoryPoint], x0: Vector8) -> Vector8:
"""Estimate the gradient of compression on the tangent space at x0.
We fit: C(γ_d(t)) ≈ C(x0) + t · <∇C, d> + O(t²)
Using linear regression across all directions and steps.
"""
C0 = points[0].compression if points else 1.0
# Collect (d, ΔC/t) pairs
samples = []
for p in points:
if p.step_size > 1e-10:
dC_dt = (p.compression - C0) / p.step_size
samples.append((p.direction, dC_dt))
if not samples:
return zero_vector(8)
# Solve least squares: ∇C ≈ argmin_Σ (d_iᵀ·g - dC/dt_i)²
# With constraint: g is tangent to S⁷ at x0 (g ⊥ x0)
D = matrix([s[0] for s in samples]) # directions as rows
y = vector([s[1] for s in samples]) # observed slopes
# Constrained least squares: minimize ||Dg - y||² s.t. g·x0 = 0
# Solution via Lagrange multipliers
g_unconstrained = D.pseudoinverse() @ y
g = g_unconstrained - dot(g_unconstrained, x0) * x0 # project to tangent
return g
4.5 SELF_ENCODE Phase (The Strange Loop)
def self_encode(state: SFFLState) -> SFFLState:
"""Encode the search trajectory as a new spiral index.
This is the KEY STRANGE LOOP: the search history becomes the next state.
The trajectory T_k = {(d_i, t_j, C_ij)} is encoded as a point on S⁷
by treating it as a probability distribution over the Hachimoji states.
CONTAINMENT guarantees (no infinite regress):
1. Trajectory is BOUNDED: finite number of points (N × M max)
2. Encoding is CONTRACTIVE: C(n_exp) ≤ max(C_history) guaranteed
3. Depth is CAPPED: max_self_ref_depth = D, then reset
"""
config = state.meta.config
trajectory = state.trajectory.history
if not trajectory:
# No history yet — skip self-encoding
state.trajectory.encoded_trajectory = state.current_point.n_spiral
return state
# ── CONTAINMENT CHECK 1: Depth cap ──────────────────────
if state.trajectory.self_ref_depth >= config.max_self_ref_depth:
# Reset: use the best point found, not the trajectory encoding
state.trajectory.self_ref_depth = 0
state.trajectory.encoded_trajectory = state.convergence.best_n
return state
# ── CONTAINMENT CHECK 2: Trajectory budget ──────────────
if len(trajectory) > config.trajectory_budget:
# Force compress before encoding
meta_point = encode_history_summary(trajectory[:-config.trajectory_budget])
trajectory = [meta_point] + trajectory[-config.trajectory_budget:]
state.trajectory.history = trajectory
# ── ENCODE TRAJECTORY ───────────────────────────────────
# Method: Treat the trajectory as an empirical distribution.
# Each trajectory point has a spiral index n_ij.
# The "distribution" of spiral indices defines a point on Δ₇.
# Step 1: Extract spiral indices from trajectory
indices = [p.spiral_index for p in trajectory]
compressions = [p.compression for p in trajectory]
# Step 2: Weight by compression (better compressions count more)
weights = softmax(compressions) # normalization
# Step 3: Compute weighted histogram on 8 bins (Hachimoji)
# Map each spiral index to a Hachimoji state via hash
histogram = [0.0] * 8
for n, w in zip(indices, weights):
h = hash_to_hachimoji(n) # deterministic: n → {0..7}
histogram[h] += w
# Normalize to probability distribution
total = sum(histogram)
p_traj = [h / total for h in histogram] # ∈ Δ₇
# Step 4: Convert to S⁷ coordinates
x_traj = [sqrt(p) for p in p_traj] # on S⁷: ||x||² = Σp = 1 ✓
# Step 5: Find closest spiral point
n_exp = spiral_index(x_traj)
# Step 6: Measure compression of the self-encoding
C_exp = compression_ratio(n_exp)
# ── CONTAINMENT CHECK 3: Contractiveness ────────────────
# The self-encoding must NOT have worse compression than
# the best point we've found. If it does, use the best point.
if C_exp < state.convergence.best_C * config.trajectory_compression_target:
# Self-encoding is too inefficient — use best point instead
n_exp = state.convergence.best_n
C_exp = state.convergence.best_C
# Update state
state.trajectory.encoded_trajectory = n_exp
state.trajectory.trajectory_compression = C_exp
state.trajectory.self_ref_depth += 1
# Record the self-reference event
self_ref_record = SelfReferenceRecord(
iteration=state.convergence.iteration,
depth=state.trajectory.self_ref_depth,
n_exp=n_exp,
C_exp=C_exp,
n_trajectory_points=len(trajectory),
)
return state
def encode_history_summary(points: List[TrajectoryPoint]) -> TrajectoryPoint:
"""Compress a set of trajectory points into a single meta-point.
This is lossy compression of the search history — it keeps enough
information to guide future search but discards individual details."""
if not points:
return None
# Compute summary statistics
avg_compression = mean(p.compression for p in points)
max_compression = max(p.compression for p in points)
avg_n = mean(p.spiral_index for p in points)
# Create a single "representative" point
return TrajectoryPoint(
iteration=points[0].iteration,
direction_index=-1, # meta-point marker
direction=zero_vector(8),
step_index=-1,
step_size=0.0,
point=points[0].point, # use first point's position
spiral_index=int(avg_n),
compression=avg_compression,
dna_encoding="META",
encoding_size_bytes=0,
famm_gate_result="META",
timestamp=points[0].timestamp,
)
4.6 META_DECIDE Phase
def meta_decide(state: SFFLState, S_star: Vector8, C_star: float)
-> Tuple[Decision, SFFLState]:
"""Decide the next state based on search results.
Three outcomes:
1. ASCEND: C* > C_k → move to S* (follow the gradient)
2. STAY: C* ≤ C_k but exploration may help → move to S_self (trajectory encoding)
3. CONVERGE: no improvement for K iterations → local maximum
"""
config = state.meta.config
k = state.convergence.iteration
C_k = state.current_point.compression
n_exp = state.trajectory.encoded_trajectory
# ── DECISION LOGIC ──────────────────────────────────────
if C_star > C_k * (1 + config.epsilon_plateau):
# Significant improvement: ASCEND
decision = Decision.ASCEND
S_next = S_star
C_next = C_star
elif state.convergence.plateau_count >= config.K_plateau:
# Plateau detected: CONVERGE
decision = Decision.CONVERGE
S_next = state.convergence.best_point
C_next = state.convergence.best_C
elif C_star > C_k:
# Marginal improvement: still ASCEND
decision = Decision.ASCEND
S_next = S_star
C_next = C_star
else:
# No improvement: use trajectory-encoded state for exploration
# This is where the strange loop feeds back
decision = Decision.STAY
S_next = spiral_point(n_exp)
C_next = state.trajectory.trajectory_compression
# ── UPDATE STATE ─────────────────────────────────────────
state.convergence.C_history.append(C_next)
state.convergence.iteration = k + 1
state.current_point = ManifoldPoint(
p=[x**2 for x in S_next],
x_sqrt=S_next,
n_spiral=spiral_index(S_next),
geodesic_origin=state.current_point.x_sqrt,
)
# ── RADIAL UPDATE ────────────────────────────────────────
state = update_radial(state, decision, C_next)
# ── CONVERGENCE CHECKS ──────────────────────────────────
converged = check_convergence(state)
halted = (k + 1) >= config.max_iterations
meltdown = check_meltdown(state)
if meltdown:
state.meta.status = State.PANIC
elif converged:
state.meta.status = State.CONVERGED
elif halted:
state.meta.status = State.HALTED
else:
state.meta.status = State.EXPLORE
# ── CHECKPOINT ───────────────────────────────────────────
if should_checkpoint(state):
dag_checkpoint(state)
return decision, state
def update_radial(state: SFFLState, decision: Decision, C: float) -> SFFLState:
"""Update radial exploration parameters."""
config = state.meta.config
old_scale = state.radial.scale
new_n = state.current_point.n_spiral
new_scale = log2(new_n + 1) if new_n > 0 else 0.0
# Compute radial velocity
velocity = new_scale - old_scale
state.radial.radial_velocity = (
config.radial_momentum * state.radial.radial_velocity
+ (1 - config.radial_momentum) * velocity
)
state.radial.scale = new_scale
state.radial.scale_history.append(new_scale)
# Determine radial mode
if state.radial.radial_velocity > 0.1:
state.radial.radial_mode = RadialMode.OUTWARD # exploring larger n
elif state.radial.radial_velocity < -0.1:
state.radial.radial_mode = RadialMode.INWARD # exploring smaller n
else:
state.radial.radial_mode = RadialMode.OSCILLATE
return state
4.7 CONVERGENCE DETECTION
def check_convergence(state: SFFLState) -> bool:
"""Multi-criteria convergence detection.
Returns True if ANY convergence criterion is met.
"""
config = state.meta.config
C_hist = state.convergence.C_history
# Criterion 1: Plateau (no improvement for K iterations)
if len(C_hist) >= config.K_plateau + 1:
recent_deltas = [C_hist[i] - C_hist[i-1] for i in range(-config.K_plateau, 0)]
if all(abs(d) < config.epsilon_plateau for d in recent_deltas):
return True
# Criterion 2: Gradient vanishing
if state.convergence.gradient_norm_history:
recent_grad_norms = list(state.convergence.gradient_norm_history)[-config.K_plateau:]
if all(g < config.delta_gradient for g in recent_grad_norms):
return True
# Criterion 3: Oscillation (compression bounces without progress)
if len(C_hist) >= 10:
recent = list(C_hist)[-10:]
mean_C = mean(recent)
std_C = std(recent)
if std_C / mean_C < config.epsilon_plateau and state.convergence.plateau_count > 0:
return True
# Criterion 4: Scar field covers manifold
scar_coverage = state.trajectory.cumulative_scar.coverage()
if scar_coverage > 0.99:
return True # everywhere has been explored or scarred
return False
def check_meltdown(state: SFFLState) -> bool:
"""Detect unrecoverable failure via Baker-analogue.
Meltdown occurs when:
1. |Λ_t| < ε(X_t) AND Ω(X_t) = 0 (collapse with no scar — impossible)
2. FAMM gate returns REJECT (too many scars — no admissible directions)
3. Gradient is NaN or infinite
4. Compression becomes negative or zero
5. State becomes numerically invalid (probabilities don't sum to 1)
"""
# Check 1: Invalid compression
C_hist = list(state.convergence.C_history)
if any(C <= 0 or isnan(C) or isinf(C) for C in C_hist[-3:]):
return True
# Check 2: Invalid probabilities
p = state.current_point.p
if abs(sum(p) - 1.0) > 1e-6 or any(pi < -1e-10 for pi in p):
return True
# Check 3: FAMM overload (too many scars)
if state.trajectory.cumulative_scar.pressure() > famm_max_pressure(state):
return True
# Check 4: Baker-analogue violation
# The invariant |Λ| ≥ ε OR Ω > 0 should ALWAYS hold
# If it doesn't, something is fundamentally wrong
Lambda = collapse_functional_full(state)
epsilon = famm_epsilon(state)
Omega = state.trajectory.cumulative_scar.total_pressure()
if abs(Lambda) < epsilon and Omega <= 0:
return True # Invariant violated — this should never happen
# Check 5: Wall time exceeded
elapsed = now() - state.meta.start_time
if elapsed > state.meta.config.max_wall_time_seconds:
return True
return False
5. THE STRANGE LOOP — Formal Specification
5.1 What It Is
The strange loop is the self-referential mechanism where the SEARCH
becomes the SUBJECT of the search. Formally:
Let T_k = { (d_i, t_j, C_ij) : i ∈ [1,N], j ∈ [1,M] } be the trajectory
of iteration k.
Define the encoding function:
encode: Trajectory → S⁷
encode(T_k) = x_exp where:
1. Compute weighted histogram of spiral indices
2. Convert to probability distribution p_traj ∈ Δ₇
3. Map to S⁷ via √p
4. Find closest spiral point: n_exp = spiral_index(x_exp)
The strange loop is:
S_{k+1} = f(S_k, T_k)
where f chooses between:
- ASCEND: S_{k+1} = argmax C(γ_d(t)) [greedy]
- STAY: S_{k+1} = encode(T_k) [self-referential]
The key property: encode(T_k) is NOT a function of S_k alone.
It depends on the ENTIRE SEARCH PROCESS of iteration k.
5.2 Why It Doesn't Cause Infinite Regress
INFINITE REGRESS would occur if:
S_{k+1} depends on T_k
T_k depends on S_k
S_k depends on T_{k-1}
...
→ S_{k+1} depends on ALL previous states and trajectories
→ memory grows without bound
→ system collapses
CONTAINMENT prevents this via three mechanisms:
┌─────────────────────────────────────────────────────────────────────┐
│ THREE CONTAINMENT LAYERS │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ LAYER 1: BOUNDED TRAJECTORY │
│ - Max N × M points per iteration │
│ - Old history summarized (lossy compression) │
│ - Trajectory budget caps total stored points │
│ │
│ Memory per iteration: O(N·M·|TrajectoryPoint|) │
│ With N=32, M=16, |TP|≈200B: ~100KB per iteration │
│ With budget=10000: max ~2MB total │
│ │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ LAYER 2: CONTRACTIVE ENCODING │
│ - The encoding function is contractive: │
│ ||encode(T_k)|| ≤ max_j ||encode({point_j})|| │
│ - Trajectory compression C(n_exp) ≤ max C(n_ij) │
│ - Self-encoding can't be worse than the best point found │
│ - If it is worse, fall back to best point (meta_decide) │
│ │
│ This guarantees: the system NEVER moves to a state with │
│ worse compression than what it's already found. │
│ │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ LAYER 3: DEPTH CAP │
│ - max_self_ref_depth = D (default 3) │
│ - After D levels of self-reference: reset to best point │
│ - This creates a "breathing" pattern: │
│ │
│ Iter 1: S_1 → explore → T_1 → encode(T_1) = S_2 │
│ Iter 2: S_2 → explore → T_2 → encode(T_2) = S_3 │
│ Iter 3: S_3 → explore → T_3 → encode(T_3) = S_4 │
│ Iter 4: depth=3 → RESET → S_5 = best_point(S_1..S_4) │
│ Iter 5: S_5 → explore → T_5 → encode(T_5) = S_6 │
│ ... │
│ │
│ The system oscillates between self-referential deepening and │
│ greedy ascent. This is INTENTIONAL — it prevents getting │
│ trapped in a basin of self-referential states. │
│ │
└─────────────────────────────────────────────────────────────────────┘
5.3 The Loop as a Fixed-Point Iteration
The strange loop can be viewed as a fixed-point iteration:
Define: F(S) = best_point( explore_from(S) )
Then: S_{k+1} = F(S_k) [greedy ascent]
With self-encoding:
S_{k+1} = α · F(S_k) + (1-α) · encode(T(S_k))
where α = adaptive weight based on improvement history.
CONVERGENCE: The iteration converges to a fixed point S* where:
S* = F(S*) (local maximum of C on S⁷)
OR: The sequence {S_k} has a convergent subsequence (Bolzano-Weierstrass
on the compact manifold S⁷), and the limit point is a stationary point
of the compression functional.
The self-encoding term (1-α)·encode(T(S_k)) acts as:
- EXPLORATION: it pushes the state away from the current basin
- REGULARIZATION: it prevents premature convergence to shallow maxima
- MEMORY: it encodes the search structure itself, making future
searches more efficient (the system "learns how to search")
6. RADIAL EXPLORATION — "Going Full Radial"
6.1 What "Full Radial" Means
Standard exploration: walk geodesics ON the surface of S⁷.
→ Changes the probability distribution p ∈ Δ₇
→ Corresponds to "which states are likely"
Radial exploration: move ALONG the spiral's radial dimension.
→ Changes the spiral index n directly
→ Corresponds to "how DEEP is the encoding"
→ Small n = shallow (few coefficients)
→ Large n = deep (many coefficients, fine-grained)
"Full radial" means: simultaneously optimize BOTH:
1. The probability distribution (SURFACE direction on S⁷)
2. The encoding depth (RADIAL direction in spiral-index space)
6.2 Radial Mode State Machine
┌─────────────────────────────────────────────────────────────────┐
│ RADIAL MODE MACHINE │
│ │
│ ┌──────────┐ C increasing ┌──────────┐ │
│ │ OSCILL │─────────────────>│ OUTWARD │ │
│ │ (start) │ │ (deepen) │ │
│ └────┬─────┘ C decreasing └────┬─────┘ │
│ ▲ │ │
│ │ C plateau │ C improving │
│ └─────────────────────────────┘ │
│ │
│ ┌──────────┐ C decreasing ┌──────────┐ │
│ │ OSCILL │<─────────────────│ INWARD │ │
│ │ │ │(shallow) │ │
│ └──────────┘ C increasing └──────────┘ │
│ │
│ Transitions: │
│ OUTWARD → INWARD: C stops improving at large n │
│ INWARD → OUTWARD: C stops improving at small n │
│ Any → OSCILLATE: C oscillates (no clear trend) │
│ OSCILLATE → Any: clear trend emerges │
│ │
└─────────────────────────────────────────────────────────────────┘
6.3 Radial Direction Generation
def generate_full_radial_directions(state: SFFLState) -> List[Direction]:
"""Generate directions combining surface + radial exploration.
Returns a MIXED set:
- N_surface directions: geodesics on S⁷ (standard)
- N_radial directions: spiral-index changes (radial)
- The ratio adapts based on radial mode
"""
config = state.meta.config
# Adaptive ratio: more radial exploration when we're in radial mode
if state.radial.radial_mode == RadialMode.OSCILLATE:
radial_fraction = 0.25 # mostly surface exploration
else:
radial_fraction = 0.5 # equal surface + radial
N_surface = int(config.N_directions * (1 - radial_fraction))
N_radial = config.N_directions - N_surface
# Surface directions (geodesics on S⁷)
surface_dirs = generate_surface_directions(state, N_surface)
# Radial directions (spiral-index changes)
radial_dirs = generate_radial_directions(
current_n=state.current_point.n_spiral,
scales=config.radial_scales,
mode=state.radial.radial_mode,
)
# Take only top N_radial by estimated promise
radial_dirs = sort_by_promise(radial_dirs)[:N_radial]
return surface_dirs + radial_dirs
6.4 Radial Momentum
def radial_momentum_update(state: SFFLState, decision: Decision) -> SFFLState:
"""Update radial velocity with momentum.
Like gradient descent with momentum, but in spiral-index space.
The velocity carries the "inertia" of the radial exploration.
"""
μ = state.meta.config.radial_momentum # velocity decay
# Compute current velocity
current_n = state.current_point.n_spiral
previous_n = state.convergence.best_n
instant_velocity = log2((current_n + 1) / (previous_n + 1))
# Update with momentum
state.radial.radial_velocity = μ * state.radial.radial_velocity + (1 - μ) * instant_velocity
# Update mode based on velocity
if abs(state.radial.radial_velocity) < 0.05:
state.radial.radial_mode = RadialMode.OSCILLATE
elif state.radial.radial_velocity > 0:
state.radial.radial_mode = RadialMode.OUTWARD
else:
state.radial.radial_mode = RadialMode.INWARD
return state
7. CHECKPOINT / RESUME SYSTEM (DAG Integration)
7.1 Checkpoint Architecture
┌─────────────────────────────────────────────────────────────────────────────┐
│ SFFL DAG CHECKPOINT STRUCTURE │
│ │
│ Each iteration produces a DAG node: │
│ │
│ DAGNode { │
│ id: uint64, │
│ parent_id: Option<uint64>, │
│ iteration: uint32, # which SFFL iteration │
│ checkpoint_type: INIT | EXPLORE | SELF_ENCODE | RECOVER, │
│ │
│ # Manifold state │
│ point_S7: Vector8, # position on S⁷ │
│ spiral_index: uint64, # n_k │
│ compression: float, # C_k │
│ │
│ # Search state │
│ trajectory_hash: bytes32, # hash of trajectory history │
│ self_ref_depth: uint8, # current nesting depth │
│ gradient_estimate: Vector8, # ∇_d C at this point │
│ │
│ # Convergence state │
│ plateau_count: uint8, │
│ best_compression: float, # global best C │
│ best_spiral_index: uint64, # global best n │
│ │
│ # FAMM integration │
│ famm_cell_id: uint64, # FAMM cell storing this state │
│ scar_field_hash: bytes32, # accumulated scar │
│ transform: Matrix8x8, # coordinate transform at this node │
│ │
│ # Determinism │
│ rng_state: bytes, # full RNG state │
│ iteration_seed: uint64, # seed for this iteration │
│ │
│ # Metadata │
│ timestamp: uint64, │
│ wall_time_ms: uint64, │
│ receipt: Receipt, # SilverSight receipt │
│ } │
│ │
│ The DAG structure enables: │
│ - Resume from any iteration │
│ - Branch exploration (try different paths from same node) │
│ - Merge results (combine findings from different branches) │
│ - Meltdown recovery (resume from last good checkpoint) │
│ │
└─────────────────────────────────────────────────────────────────────────────┘
7.2 Checkpoint Rules
def should_checkpoint(state: SFFLState) -> bool:
"""Determine if we should checkpoint now."""
config = state.meta.config
k = state.convergence.iteration
# Check every N iterations
if k % config.checkpoint_interval_iterations == 0:
return True
# Check every N seconds
elapsed = now() - state.meta.last_checkpoint_time
if elapsed > config.checkpoint_interval_seconds:
return True
# Always checkpoint before dangerous operations
if state.meta.status == State.SELF_ENCODE:
return True
return False
def dag_checkpoint(state: SFFLState) -> DAGNode:
"""Save current state as a DAG checkpoint node."""
# 1. Store FAMM cell
famm_cell = famm_bank.store(
data=serialize_state(state),
delay=f(state.convergence.best_C), # better compression = longer delay
delayMass=Tr(compute_fisher_matrix(state)),
delayWeight=state.trajectory.cumulative_scar.coverage(),
)
# 2. Compute coordinate transform from current Fisher structure
fisher_matrix = compute_fisher_matrix(state)
eigvals, eigvecs = eigh(fisher_matrix)
transform = coordinate_transform(eigvecs, eigvals)
# 3. Create DAG node
node = DAGNode(
id=dag.next_id(),
parent_id=state.checkpoint.dag_node_id,
iteration=state.convergence.iteration,
checkpoint_type=checkpoint_type_from_state(state),
point_S7=state.current_point.x_sqrt,
spiral_index=state.current_point.n_spiral,
compression=state.convergence.C_history[-1] if state.convergence.C_history else 0,
trajectory_hash=hash_trajectory(state.trajectory.history),
self_ref_depth=state.trajectory.self_ref_depth,
gradient_estimate=state.convergence.gradient_estimate,
plateau_count=state.convergence.plateau_count,
best_compression=state.convergence.best_C,
best_spiral_index=state.convergence.best_n,
famm_cell_id=famm_cell.id,
scar_field_hash=state.trajectory.cumulative_scar.hash(),
transform=transform,
rng_state=state.determinism.rng_state,
iteration_seed=state.determinism.iteration_seed,
timestamp=tick(),
wall_time_ms=elapsed_ms(state.meta.start_time),
receipt=compile_receipt(state),
)
# 4. Insert into DAG
dag.insert(node)
# 5. Update state
state.checkpoint.dag_node_id = node.id
state.checkpoint.parent_node_id = node.parent_id
state.checkpoint.famm_cell_id = famm_cell.id
state.checkpoint.transform_chain.append(transform)
state.meta.last_checkpoint_time = now()
return node
7.3 Recovery (Resume from Checkpoint)
def recover(state: SFFLState, failure_info: FailureInfo) -> SFFLState:
"""Recover from failure by resuming from the last good checkpoint.
Integration with FAMM scar system:
- The failure region is recorded as a new scar
- Future explorations will avoid this region
- The scar accumulates pressure (frustration)
"""
# 1. Record failure as scar
failure_scar = Scar(
pressure=failure_info.severity,
mode=failure_info.failure_type,
location=state.current_point.p,
iteration=state.convergence.iteration,
)
state.trajectory.cumulative_scar.add(failure_scar)
# 2. Find last good checkpoint
last_good_node = dag.find_last_good(
current=state.checkpoint.dag_node_id,
max_lookback=10,
)
if last_good_node is None:
# No good checkpoint found — PANIC
state.meta.status = State.PANIC
return state
# 3. Load checkpoint
checkpoint = dag.load(last_good_node)
famm_cell = famm_bank.load(checkpoint.famm_cell_id)
# 4. Restore state
restored_state = deserialize_state(famm_cell.data)
# 5. Update with scar information
restored_state.trajectory.cumulative_scar = state.trajectory.cumulative_scar
restored_state.checkpoint.dag_node_id = last_good_node
# 6. Advance RNG to avoid repeating the same path
restored_state.determinism.iteration_seed = hash(
restored_state.determinism.master_seed,
state.convergence.iteration,
"recover",
failure_info.failure_type,
)
# 7. Mark as recovered
restored_state.meta.status = State.EXPLORE
# 8. Emit recovery receipt
receipt = compile_recovery_receipt(state, restored_state, failure_info)
dag.insert_recovery(receipt, parent=last_good_node)
return restored_state
7.4 Meltdown Handling
def handle_meltdown(state: SFFLState) -> None:
"""Handle unrecoverable meltdown.
The Baker-analogue invariant guarantees that meltdown is rare.
When it occurs, we:
1. Dump all scars (for post-mortem analysis)
2. Emit final receipt with failure information
3. Terminate gracefully
"""
# 1. Dump scar field
scar_dump = state.trajectory.cumulative_scar.serialize()
write_file(f"meltdown_{state.meta.experiment_id}_scars.json", scar_dump)
# 2. Emit final receipt
receipt = Receipt(
receiptID=hash(state),
expression="SELF-FINDING FEEDBACK LOOP — MELTDOWN",
finalState="Ω", # Omega — scar state
ticCount=state.convergence.iteration,
fuelUsed=elapsed_ms(state.meta.start_time),
pathCost=state.convergence.best_C,
libraryRefs=["SFFL", "FAMM", "DAG", "DNA", "Baker"],
verified=False,
meltdown=True,
meltdownReason=state.meta.status.name,
)
# 3. Final DAG node
dag.insert_meltdown(receipt, parent=state.checkpoint.dag_node_id)
# 4. Terminate
state.meta.status = State.END
8. MAIN EXPERIMENT LOOP — Full Pseudocode
def run_experiment(config: ExperimentConfig) -> ExperimentResult:
"""Run the complete Self-Finding Feedback Loop experiment.
Returns the final experiment result including:
- Best compression ratio found
- Best spiral index (the "answer")
- Full trajectory (search history)
- Convergence diagnosis
- Receipt chain
"""
# ─── PHASE 0: INITIALIZE ─────────────────────────────────
state = initialize(config)
emit_receipt(state, "INIT")
try:
while state.meta.status not in {State.CONVERGED, State.HALTED, State.PANIC, State.END}:
k = state.convergence.iteration
# ─── PHASE 1: EXPLORE ──────────────────────────────
state.meta.status = State.EXPLORE
directions, state = explore(state)
# ─── PHASE 2: EVALUATE ─────────────────────────────
state.meta.status = State.EVALUATE
points, state = evaluate(state, directions)
if not points:
# All directions failed — try to recover
state = recover(state, FailureInfo(
failure_type="NO_VALID_DIRECTIONS",
severity=0.5,
))
continue
# ─── PHASE 3: SELECT BEST ──────────────────────────
state.meta.status = State.SELECT_BEST
S_star, n_star, C_star, d_star = select_best(state, points)
# ─── PHASE 4: SELF-ENCODE (the strange loop) ───────
state.meta.status = State.SELF_ENCODE
state = self_encode(state)
# ─── PHASE 5: META-DECIDE ──────────────────────────
state.meta.status = State.META_DECIDE
decision, state = meta_decide(state, S_star, C_star)
# ─── PHASE 6: CONVERGENCE CHECK ────────────────────
# (done inside meta_decide, which updates state.meta.status)
# ─── PHASE 7: CHECKPOINT ───────────────────────────
if should_checkpoint(state):
dag_checkpoint(state)
# ─── EMIT ITERATION RECEIPT ────────────────────────
emit_receipt(state, f"ITER_{k}", decision=decision)
# ─── FINAL PHASE: REPORT ─────────────────────────────
if state.meta.status == State.CONVERGED:
result = compile_result(state, status="CONVERGED")
elif state.meta.status == State.HALTED:
result = compile_result(state, status="MAX_ITERATIONS")
elif state.meta.status == State.PANIC:
result = compile_result(state, status="MELTDOWN")
else:
result = compile_result(state, status="UNKNOWN")
emit_receipt(state, "FINAL")
return result
except Exception as e:
# Catch-all: try to recover
try:
state = recover(state, FailureInfo(
failure_type="EXCEPTION",
severity=1.0,
details=str(e),
))
# Retry (bounded — max 3 recoveries)
return run_experiment_with_retry(config, max_retries=3)
except:
handle_meltdown(state)
return compile_result(state, status="FATAL")
def compile_result(state: SFFLState, status: str) -> ExperimentResult:
"""Compile the final experiment result."""
return ExperimentResult(
experiment_id=state.meta.experiment_id,
status=status,
best_compression=state.convergence.best_C,
best_spiral_index=state.convergence.best_n,
best_point=state.convergence.best_point,
final_point=state.current_point.x_sqrt,
final_compression=state.current_point.compression,
iterations=state.convergence.iteration,
trajectory_size=len(state.trajectory.history),
max_self_ref_depth_reached=state.trajectory.self_ref_depth,
scar_coverage=state.trajectory.cumulative_scar.coverage(),
dag_nodes=dag.node_count(),
wall_time_ms=elapsed_ms(state.meta.start_time),
convergence_diagnosis=diagnose_convergence(state),
receipt_chain=dag.receipt_chain(),
)
9. INTEGRATION SPEC — FAMM + DAG + DNA
9.1 FAMM Integration
┌─────────────────────────────────────────────────────────────────────┐
│ FAMM INTEGRATION POINTS │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ 1. SCAR ACCUMULATION │
│ - Every failed exploration direction → FAMM scar │
│ - Scar pressure = |Λ_t| (collapse functional) │
│ - Scar location = point on S⁷ where failure occurred │
│ - Integration: state.trajectory.cumulative_scar │
│ │
│ 2. GATE CHECKING │
│ - Before each geodesic step: FAMM gate check │
│ - |Λ_t| ≥ ε ? → ADMIT (proceed) │
│ - |Λ_t| < ε ? → SCAR (record, skip direction) │
│ - Too many scars? → REJECT (trigger recovery) │
│ - Integration: evaluate() famm_gate_result field │
│ │
│ 3. DELAY-LINE STORAGE │
│ - Each checkpoint stored as FAMM cell │
│ - delay = f(compression) — better C = longer delay │
│ - delayMass = Tr(Fisher matrix) — total curvature │
│ - delayWeight = scar coverage — fraction of space explored │
│ - Integration: dag_checkpoint() famm_bank.store() │
│ │
│ 4. BAKER-ANALOGUE INVARIANT │
│ - Maintained: |Λ_t| ≥ ε(X_t) OR Ω(X_t) > 0 │
│ - Violation → meltdown (PANIC state) │
│ - Integration: check_meltdown() │
│ │
│ 5. FRUSTRATION AS SIGNAL │
│ - High frustration = high curvature = promising region │
│ - Frustration guides direction selection │
│ - Integration: scar_filter() prioritizes low-frustration dirs │
│ │
└─────────────────────────────────────────────────────────────────────┘
9.2 DAG Integration
┌─────────────────────────────────────────────────────────────────────┐
│ DAG INTEGRATION POINTS │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ 1. CHECKPOINT NODES │
│ - One DAG node per checkpoint iteration │
│ - Node stores: state snapshot + transform + receipt │
│ - Parent = previous checkpoint (tree structure) │
│ - Integration: dag_checkpoint() → dag.insert() │
│ │
│ 2. RESUME FROM ANY NODE │
│ - dag.resume(node_id) → checkpoint → state │
│ - FAMM cell loaded from checkpoint │
│ - RNG state restored deterministically │
│ - Integration: recover() → dag.load() │
│ │
│ 3. BRANCHING (future: parallel exploration) │
│ - Multiple children from same parent = branches │
│ - Each branch explores different region │
│ - Integration: dag.insert(parent=node_id) │
│ │
│ 4. RECEIPT CHAIN │
│ - Each checkpoint has a SilverSight receipt │
│ - Receipt chain = experiment audit log │
│ - Integration: compile_receipt() per checkpoint │
│ │
│ 5. MELTDOWN RECOVERY │
│ - dag.find_last_good() — walk back from failure │
│ - dag.insert_meltdown() — record failure │
│ - Integration: handle_meltdown(), recover() │
│ │
└─────────────────────────────────────────────────────────────────────┘
9.3 DNA Encoding Integration
┌─────────────────────────────────────────────────────────────────────┐
│ DNA ENCODING INTEGRATION │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ 1. SPIRAL INDEX → DNA │
│ - n → phinary(n) → RLE → DNA bases (A,B,C,G,P,S,T,Z) │
│ - Used for: compression measurement, checkpoint storage │
│ - Integration: compression_ratio() → phinary_encode() → RLE() │
│ │
│ 2. TRAJECTORY ENCODING (strange loop) │
│ - trajectory → weighted histogram → p ∈ Δ₇ → √p ∈ S⁷ │
│ - S⁷ point → spiral_index → n_exp → DNA │
│ - Integration: self_encode() → spiral_index() → DNA │
│ │
│ 3. STATE SERIALIZATION │
│ - Full state → DNA encoding → FAMM cell storage │
│ - Enables: checkpointing, replication, audit │
│ - Integration: serialize_state() → dna_encode() │
│ │
│ 4. RECEIPT ENCODING │
│ - Each receipt gets DNA-encoded receipt ID │
│ - Receipt chain = DNA chain (verifiable) │
│ - Integration: compile_receipt() → hash → dna_encode() │
│ │
│ 5. HACHIMOJI STATE ↔ S⁷ │
│ - Stack distribution → Δ₇ → S⁷ │
│ - DNA alphabet = 8 Hachimoji states ↔ 8 simplex vertices │
│ - Integration: current_point.p ↔ Hachimoji stack │
│ │
└─────────────────────────────────────────────────────────────────────┘
10. CONVERGENCE CRITERIA — Summary
10.1 Convergence Detection Matrix
| Criterion | Condition | Meaning | Action |
|---|---|---|---|
| Plateau | |ΔC| < ε for K iterations |
No improvement — local max | STOP (CONVERGED) |
| Gradient vanish | ||∇C|| < δ |
Flat region — no direction to go | STOP (CONVERGED) |
| Oscillation | std(C) / mean(C) < ε for 10 iters |
Bouncing without progress | STOP (CONVERGED) |
| Scar coverage | Ω.coverage() > 0.99 |
All space explored or scarred | STOP (CONVERGED) |
| Max iterations | k ≥ max_iterations |
Safety limit reached | STOP (HALTED) |
| Wall time | elapsed > max_wall_time |
Hard timeout | STOP (HALTED) |
| Meltdown | Baker-analogue violated | System failure | PANIC |
10.2 Convergence Receipt
{
"receiptID": "sha256(experiment_result)",
"expression": "SELF-FINDING FEEDBACK LOOP — Radial Self-Finding Experiment",
"finalState": "Φ",
"ticCount": 42,
"fuelUsed": 1234567,
"pathCost": 1048576.0,
"bestCompression": 1048576.0,
"bestSpiralIndex": 3141592653589,
"iterations": 42,
"convergenceType": "PLATEAU",
"selfRefMaxDepth": 3,
"scarCoverage": 0.23,
"dagNodes": 5,
"libraryRefs": ["SFFL", "FAMM", "DAG", "DNA", "Metric", "Baker", "Chunk"],
"verified": true,
"identityCheck": "state.Introspect == expected",
"receiptChain": ["init_receipt", "iter_10", "iter_20", "iter_30", "iter_40", "final"]
}
11. DETERMINISM GUARANTEE
11.1 Seeding Hierarchy
master_seed (64-bit, user-configurable, default: 0xFEEDFACE42424242)
│
├── iteration_seed(k) = hash(master_seed, k, "iteration")
│ └── Used for: state initialization at iteration k
│
├── direction_seed(k) = hash(master_seed, k, "directions")
│ └── Used for: generating N directions at iteration k
│
├── step_seed(k) = hash(master_seed, k, "steps")
│ └── Used for: step-size sampling along geodesics
│
└── recovery_seed(k, attempt) = hash(master_seed, k, "recover", attempt)
└── Used for: RNG after recovery (different path)
11.2 Determinism Checklist
| Source of Non-Determinism | Our Fix |
|---|---|
| Random number generation | Seeded hierarchy (above) |
| Hash ordering | Sort all collections before encoding |
| Floating-point | Q16.16 fixed-point for all stored values |
| Memory addresses | Encode logical structure, not addresses |
| Timing | Snapshot state, don't encode timing |
| Parallel execution | Deterministic scheduling (round-robin) |
| OS differences | Pure computation, no OS calls |
11.3 Reproducibility Proof Sketch
Theorem: The SFFL experiment is fully reproducible.
Proof:
Given: same master_seed, same config, same code
Then:
1. All RNG sequences are identical (seeded hierarchy)
2. All direction generations are identical
3. All geodesic walks follow the same path
4. All compression measurements are identical (fixed-point)
5. All FAMM gate decisions are identical
6. All state transitions follow the same path
Therefore: The entire experiment trace is deterministic.
Corollary: Two runs with the same seed produce identical:
- Trajectory history
- Convergence point
- Receipt chain
- DAG structure
- Final result
12. STATE MACHINE DIAGRAM (ASCII)
┌─────────────┐
│ IDLE │
└──────┬──────┘
│ init()
▼
┌─────────────┐
┌────────────────────────>│ INIT │
│ (recover resume) └──────┬──────┘
│ │
│ ┌───────────────────────────┘
│ │
│ ▼ ┌──────────┐
│ ┌──────────┐ meltdown │ PANIC │
│ │ EXPLORE │────────────────>│ │
│ └────┬─────┘ │ (unrecov)│
│ │ directions └────┬─────┘
│ │ generated │
│ ▼ │ scar_dump
│ ┌──────────┐ ▼
│ │ EVALUATE │ ┌──────────┐
│ └────┬─────┘ │ END │
│ │ compression └──────────┘
│ │ measured ▲
│ ▼ │
│ ┌──────────┐ plateau × K ┌──────────┐
│ │ SELECT │────────────────>│ CONVERGED│
│ │ BEST │ │ │
│ └────┬─────┘ │ report() │
│ │ best found └────┬─────┘
│ ▼ │
│ ┌──────────┐ │
│ │ SELF- │ │
│ │ ENCODE │ │
│ └────┬─────┘ │
│ │ trajectory │
│ │ encoded │
│ ▼ │
│ ┌──────────┐ max_iter ┌──────────┐
│ │ META │────────────────>│ HALTED │
│ │ DECIDE │ │ │
│ └────┬─────┘ │ report() │
│ │ decision └────┬─────┘
│ │ made │
│ └─────────────────────────────┘
│ (report → END)
│
└─────── (convergence check: if not converged, loop back)
Any state ──failure──> RECOVER ──success──> [previous state]
RECOVER ──fail────> PANIC
13. THE COMPLETE UPDATE EQUATIONS
13.1 Manifold Position Update
x_{k+1} = { γ_{d*}(t*) if ASCEND (follow best direction)
{ x_exp if STAY (self-encoded trajectory)
{ x_best if CONVERGE (best point overall)
where:
d* = argmax_{d_i} max_j C(spiral_index(γ_{d_i}(t_j)))
t* = argmax_j C(spiral_index(γ_{d*}(t_j)))
x_exp = √p_traj where p_traj = weighted_histogram(trajectory)
x_best = argmax_{x ∈ {all explored}} C(spiral_index(x))
13.2 Compression Update
C_{k+1} = C(spiral_index(x_{k+1}))
C_best = max(C_best, C_{k+1})
plateau_count = { 0 if C_{k+1} > C_k + ε
{ plateau_count + 1 otherwise
13.3 Trajectory Update
T_{k+1} = T_k ∪ { (d_i, t_j, C_ij) : i∈[1,N], j∈[1,M] }
if |T_{k+1}| > budget:
T_{k+1} = { encode_summary(T_k[:-budget]) } ∪ T_k[-budget:]
n_exp = spiral_index( encode(T_{k+1}) )
depth_{k+1} = { depth_k + 1 if n_exp ≠ n_k
{ 0 if depth_k ≥ D
13.4 Scar Field Update
Ω_{k+1} = Ω_k + Σ_{failed explorations} Scar(pressure=|Λ|, location=x)
where Λ = collapse_functional(state, x) at each failed point
13.5 Radial State Update
v_{k+1} = μ · v_k + (1-μ) · (log₂(n_{k+1}+1) - log₂(n_k+1))
mode_{k+1} = { OUTWARD if v_{k+1} > 0.1
{ INWARD if v_{k+1} < -0.1
{ OSCILLATE otherwise
13.6 Checkpoint Update
node_{k+1} = DAGNode(
parent = node_k,
point = x_{k+1},
compression = C_{k+1},
transform = eigenstructure_transform(Fisher(x_{k+1})),
rng_state = rng.snapshot(),
)
14. APPENDIX: GLOSSARY
| Term | Meaning |
|---|---|
| SFFL | Self-Finding Feedback Loop (this system) |
| S⁷ | 7-sphere (Fisher information manifold in √p-coordinates) |
| Δ₇ | 7-simplex (probability distributions over 8 states) |
| Φ-corkscrew | Spiral f(n) = (√n·cos(nψ), √n·sin(nψ)) with ψ = 2π/Φ² |
| spiral_index | Map from S⁷ point to closest spiral point index |
| C(n) | Compression ratio = original_size / RLE(phinary(n))_size |
| strange loop | Search trajectory becomes the next search's state |
| self_ref_depth | Nesting level of self-reference (capped at D) |
| scar | Recorded failure region on the manifold (FAMM) |
| Baker-analogue | Invariant: |Λ| ≥ ε OR Ω > 0 (no silent failures) |
| FAMM | Frustrated Access Memory Module (delay-line memory) |
| DAG | Directed Acyclic Graph of checkpoints |
| FSDU | FAMM Scar Differential Update |
| Hachimoji | 8-symbol DNA alphabet: A,B,C,G,P,S,T,Z ↔ Φ,Λ,Ρ,Κ,Ω,Σ,Π,Ζ |
Design completed. Ready for implementation.
Version: 1.0 Date: 2025-06-23 System: SFFL v1 — Radial Self-Finding Experiment